Solve the following equations :
`(1)/(x-1)+(2)/(x-2)=(3)/(x-3)`

Answer

To solve the equation \(\frac{1}{x-1} + \frac{2}{x-2} = \frac{3}{x-3}\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the fractions is \((x-1)(x-2)(x-3)\). We will multiply each term by this common denominator to eliminate the fractions.

\[ (x-1)(x-2)(x-3) \left(\frac{1}{x-1} + \frac{2}{x-2} = \frac{3}{x-3}\right) \]

### Step 2: Distribute the common denominator
Distributing the common denominator to each term gives us:

\[ (x-2)(x-3) + 2(x-1)(x-3) = 3(x-1)(x-2) \]

### Step 3: Expand each term
Now we will expand each of the terms:

1. \((x-2)(x-3) = x^2 - 5x + 6\)
2. \(2(x-1)(x-3) = 2(x^2 - 4x + 3) = 2x^2 - 8x + 6\)
3. \(3(x-1)(x-2) = 3(x^2 - 3x + 2) = 3x^2 - 9x + 6\)

Putting it all together, we have:

\[ x^2 - 5x + 6 + 2x^2 - 8x + 6 = 3x^2 - 9x + 6 \]

### Step 4: Combine like terms
Combine the left-hand side:

\[ (1x^2 + 2x^2) + (-5x - 8x) + (6 + 6) = 3x^2 - 13x + 12 \]

So, the equation now looks like:

\[ 3x^2 - 13x + 12 = 3x^2 - 9x + 6 \]

### Step 5: Move all terms to one side
Subtract \(3x^2\) from both sides:

\[ -13x + 12 = -9x + 6 \]

Now, add \(9x\) to both sides:

\[ -4x + 12 = 6 \]

### Step 6: Isolate \(x\)
Subtract \(12\) from both sides:

\[ -4x = 6 - 12 \]

This simplifies to:

\[ -4x = -6 \]

Now, divide by \(-4\):

\[ x = \frac{-6}{-4} = \frac{3}{2} \]

### Final Answer
Thus, the solution to the equation is:

\[ x = \frac{3}{2} \]